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<h1>Simple Dome Model</h1>

<p>Now let's have a closer look at the geometry. In figure 44, the dotted line througth the points P1, P4 and P2 represents the surface of the plantilla, and serves as our reference line (or plane in 3D). The hatched area is a cross section of the carved solera at the body centerline. </p>

<p class="imgbox"><a name="fig-044"></a><img src="../img/fig-044.png" style="max-width: 400px;"/><br />Figure 44: Cross-section through the body centerline of the carved solera.</p>

<h2>Definition of Names</h2>

<p>The carved dome starts P0 at the body top, and ends at P2 at the body bottom. The curved line through P1, P3 and P2 will force the soundboard into the desired shape. The points are located on a circle around P0 at dome radius DR:</p>

<pre class="codebox formula">
P0P1 = P0P2 = P0P3 = DR
</pre>

<p>The body length BL is the distance between P1 and P2:</p>

<pre class="codebox formula">
P1P2 = BL
</pre>

<p>The center of the dome is at P3, and the line through the circle center <span class="overline">P0P3</span> divides <span class="overline">P1P2</span> into equal halves at P4, which is half the body length BL/2:</p>

<pre class="codebox formula">
P1P4 = P4P2 = P1P2 / 2 = BL / 2
</pre>

<p>The maximum dome height DH is at the center of the dome below P4:</p>

<pre class="codebox formula">
P3P4 = DH@P4 = DH
</pre>

<p>The line through the body end points P1 P2 is the <em>reference line</em> for measurements. This line also represents a flat soundboard without dome.</p>

<h2>Dome Height</h2>

<p>Let's start out with calculating the maximum dome height DH.</p>

<p>The lines <span class="overline">P0P3</span> and <span class="overline">P1P2</span> are crossing at a right angle, which allows us to apply the pythagorean theorem on the triangle P0 P4 P3:</p>

<pre class="codebox formula">
P0P4<sup>2</sup> + P4P1<sup>2</sup> = P1P0<sup>2</sup>
</pre>

<p><span class="overline">P0P1</span> and <span class="overline">P1P4</span> are known, so we develop for <span class="overline">P0P4</span>:</p>

<pre class="codebox formula">
P0P4<sup>2</sup> = P1P0<sup>2</sup> &minus; P1P4<sup>2</sup>
P0P4 = sqrt( P1P0<sup>2</sup> &minus; P1P4<sup>2</sup> )
</pre>

<p><span class="overline">P0P3</span> is the sum of <span class="overline">P0P4</span> and <span class="overline">P4P3</span>. We develop that for P4P3:</p>

<pre class="codebox formula">
P0P4 + P3P4 + P0P3
P3P4 = P0P3 &minus; P0P4
</pre>

<p>An expression for P0P4 has been derived above, so we can substitute that:</p>

<pre class="codebox formula">
P0P4 = sqrt( P1P0<sup>2</sup> &minus; P1P4<sup>2</sup> )
P3P4 = P0P3 &minus; sqrt( P1P0<sup>2</sup> &minus; P1P4<sup>2</sup> )
</pre>

<p>We are nearly done. The last steps is to substitute the the abstract lines with the more familiar expressions:</p>

<pre class="codebox formula">
P3P4 = DH
P0P3 = P0P1 = DR
P1P4 = BL / 2
DHmax = DH@P4 = DR &minus; sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> )
</pre>

<p>Which give us the final formula for the maximum dome height:</p>

<pre class="codebox formula">
DH = DH@P4 = DR &minus; sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> )
</pre>

<h2>Dome Radius</h2>

<p>Next step is to provide a formula to calculate the dome radius from the other values, body length BL and dome height DH. We simple take the formula for the maximum dome heigh DH, and develop for DR:</p>

<pre class="codebox formula">
DH + sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> ) = DR
DH = DR &minus; sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> )
</pre>

<p>That was simple, wasn't it?</p>

<h2>Body Length</h2>

<p>The formula for the body length BL is developed similarly.</p>

<pre class="codebox formula">
DH + sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> ) = DR
sqrt( DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> ) = DR &minus; DH 
DR<sup>2</sup> &minus; ( BL / 2 )<sup>2</sup> = ( DR &minus; DH )<sup>2</sup>
( BL / 2 )<sup>2</sup> =  DR<sup>2</sup> &minus; ( DR &minus; DH )<sup>2</sup>
BL / 2 =  sqrt( DR<sup>2</sup> &minus; ( DR &minus; DH )<sup>2</sup> ) 
</pre>

<p>The last step gives us the formula the body length BL:</p>

<pre class="codebox formula">
BL =  2 &times; sqrt( DR<sup>2</sup> &minus; ( DR &minus; DHmax )<sup>2</sup> )
</pre>

<h2>Maximum Dome Height Calculator</h2>

<p>I have built the formulas into a calculator, which allows us to easily play with different combinations of values.</p>

<iframe class="calculator" frameborder="0" scrolling="no" src="../calculators/simple-dome-frame.htm" 
	style="width: 100%; " onload="this.style.height = this.contentWindow.document.body.scrollHeight + 'px'">
	</iframe>

<p>Alternatively, you can use the calulator in a <a href="javascript:openCalculatorWindow( '../calculators/', 'simple-dome-frame.htm', 'simpleDomeCalculatorWindow', 400, 220 )">separate window</a>. You might also find the <a href="javascript:openCalculatorWindow( '../calculators/', 'imperial-metric-frame.htm', 'imperialMetricConverterWindow', 400, 220 )">Imperial metric converter</a> handy.</p>

<h2>Conclusion</h2>

<p>If you have played with the model for a while, you may have noticed that the calculated dome height is somewhat high, compared to the usual values in the literature or to the pulications of various builders. And there is another problem: The dome center <em>is not at the bridge position</em>. But for a correct layout of the guitar geometry, we need to know the dome height at exactly that point.</p>

<p>This calls for a refined model. See you on the next page.</p>


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